68 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
New bounds for odd colourings of graphs
Given a graph , a vertex-colouring of , and a subset
, a colour is said to be \emph{odd} for
in if it has an odd number of occurrences in . We say that
is an \emph{odd colouring} of if it is proper and every (open)
neighbourhood has an odd colour in . The odd chromatic number of a
graph , denoted by , is the minimum such that an
odd colouring exists. In a recent paper, Caro,
Petru\v sevski and \v Skrekovski conjectured that every connected graph of
maximum degree has odd-chromatic number at most . We
prove that this conjecture holds asymptotically: for every connected graph
with maximum degree , as . We also prove that for every
. If moreover the minimum degree of is sufficiently large,
we have and . Finally, given an integer , we study the
generalisation of these results to -odd colourings, where every vertex
must have at least odd colours in its neighbourhood. Many
of our results are tight up to some multiplicative constant
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvistâs B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Uniformly Random Colourings of Sparse Graphs
We analyse uniformly random proper -colourings of sparse graphs with
maximum degree in the regime . This regime
corresponds to the lower side of the shattering threshold for random graph
colouring, a paradigmatic example of the shattering threshold for random
Constraint Satisfaction Problems. We prove a variety of results about the
solution space geometry of colourings of fixed graphs, generalising work of
Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the
performance of stochastic local search algorithms in this regime. Our central
proof relies only on elementary techniques, namely the first-moment method and
a quantitative induction, yet it strengthens list-colouring results due to Vu,
and more recently Davies, Kang, P., and Sereni, and generalises
state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It
further yields an approximately tight lower bound on the number of colourings,
also known as the partition function of the Potts model, with implications for
efficient approximate counting
Linearly ordered colourings of hypergraphs
A linearly ordered (LO) -colouring of an -uniform hypergraph assigns an
integer from to every vertex so that, in every edge, the
(multi)set of colours has a unique maximum. Equivalently, for , if two
vertices in an edge are assigned the same colour, then the third vertex is
assigned a larger colour (as opposed to a different colour, as in classic
non-monochromatic colouring). Barto, Battistelli, and Berg [STACS'21] studied
LO colourings on -uniform hypergraphs in the context of promise constraint
satisfaction problems (PCSPs). We show two results.
First, given a 3-uniform hypergraph that admits an LO -colouring, one can
find in polynomial time an LO -colouring with .
Second, given an -uniform hypergraph that admits an LO -colouring, we
establish NP-hardness of finding an LO -colouring for every constant
uniformity . In fact, we determine relationships between
polymorphism minions for all uniformities , which reveals a key
difference between and and which may be of independent
interest. Using the algebraic approach to PCSPs, we actually show a more
general result establishing NP-hardness of finding an LO -colouring for LO
-colourable -uniform hypergraphs for and .Comment: Full version (with stronger both tractability and intractability
results) of an ICALP 2022 pape
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Trois résultats en théorie des graphes
Cette thĂšse rĂ©unit en trois articles mon intĂ©rĂȘt Ă©clectique pour la thĂ©orie des graphes.
Le premier problÚme étudié est la conjecture de Erdos-Faber-Lovåsz:
La rĂ©union de k graphes complets distincts, ayant chacun k sommets, qui ont deux-Ă -deux au plus un sommet en commun peut ĂȘtre proprement coloriĂ©e en k couleurs.
Cette conjecture se caractĂ©rise par le peu de rĂ©sultats publiĂ©s. Nous prouvons quâune nouvelle classe de graphes, construite de maniĂšre inductive, satisfait la conjecture. Le rĂ©sultat consistera Ă prĂ©senter une classe qui ne prĂ©sente pas les limitations courantes dâuniformitĂ© ou de rĂ©gularitĂ©.
Le deuxiĂšme problĂšme considĂšre une conjecture concernant la couverture des arĂȘtes dâun graphe:
Si G est un graphe simple avec alpha(G) = 2, alors le nombre minimum de cliques nĂ©cessaires pour couvrir lâensemble des arĂȘtes de G (notĂ© ecc(G)) est au plus n, le nombre de sommets de G.
La meilleure borne connue satisfaite par ecc(G) pour tous les graphes avec nombre dâindĂ©pendance de deux est le minimum de n + delta(G) et 2n â omega(racine (n log n)), oĂč delta(G) est le plus petit nombre de voisins dâun sommet de G. Notre objectif a Ă©tĂ© dâobtenir la borne ecc(G) <= 3/2 n pour une classe de graphes la plus large possible. Un autre rĂ©sultat associĂ© Ă ce problĂšme apporte la preuve de la conjecture pour une classe particuliĂšre de graphes:
Soit G un graphe simple avec alpha(G) = 2. Si G a une arĂȘte dominante uv telle que G \ {u,v} est de diamĂštre 3, alors ecc(G) <= n.
Le troisiĂšme problĂšme Ă©tudie le jeu de policier et voleur sur un graphe. Presque toutes les Ă©tudes concernent les graphes statiques, et nous souhaitons explorer ce jeu sur les graphes dynamiques, dont les ensembles dâarĂȘtes changent au cours du temps. Nowakowski et Winkler caractĂ©risent les graphes statiques pour lesquels un unique policier peut toujours attraper
le voleur, appellĂ©s cop-win, Ă lâaide dâune relation <= dĂ©finie sur les sommets de ce graphe:
Un graphe G est cop-win si et seulement si la relation <= définie sur ses sommets est triviale.
Nous adaptons ce thĂ©orĂšme aux graphes dynamiques. Notre dĂ©marche nous mĂšne Ă une relation nous permettant de prĂ©senter une caractĂ©risation des graphes dynamiques cop-win. Nous donnons ensuite des rĂ©sultats plus spĂ©cifiques aux graphes pĂ©riodiques. Nous indiquons aussi comment gĂ©nĂ©raliser nos rĂ©sultats pour k policiers et l voleurs en rĂ©duisant ce cas Ă celui dâun policier unique et un voleur unique. Un algorithme pour dĂ©cider si, sur un graphe pĂ©riodique donnĂ©, k policiers peuvent capturer l voleurs dĂ©coule de notre caractĂ©risation.This thesis represents in three articles my eclectic interest for graph theory.
The first problem is the conjecture of Erdos-Faber-LovĂĄsz:
If k complete graphs, each having k vertices, have the property that every pair of distinct complete graphs have at most one vertex in common, then the vertices of the resulting graph can be properly coloured by using k colours.
This conjecture is notable in that only a handful of classes of EFL graphs are proved to satisfy the conjecture. We prove that the Erdos-Faber-LovĂĄsz Conjecture holds for a new class of graphs, and we do so by an inductive argument. Furthermore, graphs in this class have no restrictions on the number of complete graphs to which a vertex belongs or on the
number of vertices of a certain type that a complete graph must contain.
The second problem addresses a conjecture concerning the covering of the edges of a graph:
The minimal number of cliques necessary to cover all the edges of a simple graph G is denoted by ecc(G). If alpha(G) = 2, then ecc(G) <= n.
The best known bound satisfied by ecc(G) for all the graphs with independence number two is the minimum of n + delta(G) and 2n â omega(square root (n log n)), where delta(G) is the smallest number of neighbours of a vertex in G.
In this type of graph, all the vertices at distance two from a given vertex form a clique. Our approach is to extend all of these n cliques in order to cover the maximum possible number of edges. Unfortunately, there are graphs for which itâs impossible to cover all the edges with this method. However, we are able to use this approach to prove a bound of ecc(G) <= 3/2n for some newly studied infinite families of graphs.
The third problem addresses the game of Cops and Robbers on a graph. Almost all the articles concern static graphs, and we would like to explore this game on dynamic graphs, the edge sets of which change as a function of time. Nowakowski and Winkler characterize static graphs for which a cop can always catch the robber, called cop-win graphs, by means of a relation <= defined on the vertices of such graphs:
A graph G is cop-win if and only if the relation <= defined on its vertices is trivial.
We adapt this theorem to dynamic graphs. Our approach leads to a relation, that allows us to present a characterization of cop-win dynamic graphs. We will then give more specific results for periodic graphs, and we will also indicate how to generalize our results to k cops and l robbers by reducing this case to one with a single cop and a single robber. An
algorithm to decide whether on a given periodic graph k cops can catch l robbers follows from our characterization
Proper Conflict-free Coloring of Graphs with Large Maximum Degree
A proper coloring of a graph is conflict-free if, for every non-isolated
vertex, some color is used exactly once on its neighborhood. Caro,
Petru\v{s}evski, and \v{S}krekovski proved that every graph has a proper
conflict-free coloring with at most colors and conjectured that
colors suffice for every connected graph with . Our first main result is that even for list-coloring, colors suffice for every graph
with ; we also prove slightly weaker bounds for all
graphs with . These results follow from our more general
framework on proper conflict-free list-coloring of a pair consisting of a graph
and a ``conflict'' hypergraph . As another corollary of our
results in this general framework, every graph has a proper
-list-coloring such that every bi-chromatic
component is a path on at most three vertices, where the number of colors is
optimal up to a constant factor. Our proof uses a fairly new type of recursive
counting argument called Rosenfeld counting, which is a variant of the
Lov\'{a}sz Local Lemma or entropy compression.
We also prove an asymptotically optimal result for a fractional analogue of
our general framework for proper conflict-free coloring for pairs of a graph
and a conflict hypergraph. A corollary states that every graph has a
fractional -coloring such that every fractionally
bi-chromatic component has at most two vertices. In particular, it implies that
the fractional analogue of the conjecture of Caro et al. holds asymptotically
in a strong sense
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
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